Theorem ismkvnex 7119

Description: The predicate of being Markov stated in terms of double negation and comparison with 1_o. (Contributed by Jim Kingdon, 29-Nov-2023.)

Assertion

Ref	Expression
ismkvnex	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))

Distinct variable groups:   𝐴,𝑓,𝑥   𝑓,𝑉,𝑥

Proof of Theorem ismkvnex

Dummy variables 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5485	. . . . . . . . 9 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))) → (𝑔‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥))
2	1	eqeq1d 2174	. . . . . . . 8 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))) → ((𝑔‘𝑥) = 1o ↔ ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = 1o))
3	2	ralbidv 2466	. . . . . . 7 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = 1o))
4	3	notbid 657	. . . . . 6 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))) → (¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o ↔ ¬ ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = 1o))
5	1	eqeq1d 2174	. . . . . . 7 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))) → ((𝑔‘𝑥) = ∅ ↔ ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = ∅))
6	5	rexbidv 2467	. . . . . 6 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))) → (∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = ∅))
7	4, 6	imbi12d 233	. . . . 5 ⊢ (𝑔 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))) → ((¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅) ↔ (¬ ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = ∅)))
8		elex 2737	. . . . . . 7 ⊢ (𝐴 ∈ Markov → 𝐴 ∈ V)
9		ismkvmap 7118	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 ∈ Markov ↔ ∀𝑔 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅)))
10	9	biimpd 143	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∈ Markov → ∀𝑔 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅)))
11	8, 10	mpcom 36	. . . . . 6 ⊢ (𝐴 ∈ Markov → ∀𝑔 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅))
12	11	adantr 274	. . . . 5 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → ∀𝑔 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅))
13		elmapi 6636	. . . . . . . . . 10 ⊢ (𝑓 ∈ (2o ↑𝑚 𝐴) → 𝑓:𝐴⟶2o)
14	13	adantl 275	. . . . . . . . 9 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → 𝑓:𝐴⟶2o)
15	14	ffvelrnda 5620	. . . . . . . 8 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑓‘𝑧) ∈ 2o)
16		2oconcl 6407	. . . . . . . 8 ⊢ ((𝑓‘𝑧) ∈ 2o → (1o ∖ (𝑓‘𝑧)) ∈ 2o)
17	15, 16	syl 14	. . . . . . 7 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (1o ∖ (𝑓‘𝑧)) ∈ 2o)
18	17	fmpttd 5640	. . . . . 6 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))):𝐴⟶2o)
19		2onn 6489	. . . . . . . 8 ⊢ 2o ∈ ω
20	19	a1i 9	. . . . . . 7 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → 2o ∈ ω)
21		simpl 108	. . . . . . 7 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → 𝐴 ∈ Markov)
22	20, 21	elmapd 6628	. . . . . 6 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))) ∈ (2o ↑𝑚 𝐴) ↔ (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))):𝐴⟶2o))
23	18, 22	mpbird 166	. . . . 5 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))) ∈ (2o ↑𝑚 𝐴))
24	7, 12, 23	rspcdva 2835	. . . 4 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → (¬ ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = ∅))
25		eqid 2165	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧))) = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))
26		fveq2 5486	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑓‘𝑧) = (𝑓‘𝑥))
27	26	difeq2d 3240	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (1o ∖ (𝑓‘𝑧)) = (1o ∖ (𝑓‘𝑥)))
28		simpr 109	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
29		1oex 6392	. . . . . . . . . . 11 ⊢ 1o ∈ V
30		difexg 4123	. . . . . . . . . . 11 ⊢ (1o ∈ V → (1o ∖ (𝑓‘𝑥)) ∈ V)
31	29, 30	mp1i 10	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (1o ∖ (𝑓‘𝑥)) ∈ V)
32	25, 27, 28, 31	fvmptd3 5579	. . . . . . . . 9 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = (1o ∖ (𝑓‘𝑥)))
33	32	eqeq1d 2174	. . . . . . . 8 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = 1o ↔ (1o ∖ (𝑓‘𝑥)) = 1o))
34		difeq2 3234	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) = ∅ → (1o ∖ (𝑓‘𝑥)) = (1o ∖ ∅))
35		dif0 3479	. . . . . . . . . . . 12 ⊢ (1o ∖ ∅) = 1o
36	34, 35	eqtrdi 2215	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) = ∅ → (1o ∖ (𝑓‘𝑥)) = 1o)
37	36	adantl 275	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = ∅) → (1o ∖ (𝑓‘𝑥)) = 1o)
38		1n0 6400	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
39	38	nesymi 2382	. . . . . . . . . . . 12 ⊢ ¬ ∅ = 1o
40		eqeq1 2172	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) = ∅ → ((𝑓‘𝑥) = 1o ↔ ∅ = 1o))
41	39, 40	mtbiri 665	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) = ∅ → ¬ (𝑓‘𝑥) = 1o)
42	41	adantl 275	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = ∅) → ¬ (𝑓‘𝑥) = 1o)
43	37, 42	2thd 174	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = ∅) → ((1o ∖ (𝑓‘𝑥)) = 1o ↔ ¬ (𝑓‘𝑥) = 1o))
44		difid 3477	. . . . . . . . . . . . . 14 ⊢ (1o ∖ 1o) = ∅
45	44	eqeq1i 2173	. . . . . . . . . . . . 13 ⊢ ((1o ∖ 1o) = 1o ↔ ∅ = 1o)
46	39, 45	mtbir 661	. . . . . . . . . . . 12 ⊢ ¬ (1o ∖ 1o) = 1o
47		difeq2 3234	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑥) = 1o → (1o ∖ (𝑓‘𝑥)) = (1o ∖ 1o))
48	47	eqeq1d 2174	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) = 1o → ((1o ∖ (𝑓‘𝑥)) = 1o ↔ (1o ∖ 1o) = 1o))
49	46, 48	mtbiri 665	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) = 1o → ¬ (1o ∖ (𝑓‘𝑥)) = 1o)
50	49	adantl 275	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = 1o) → ¬ (1o ∖ (𝑓‘𝑥)) = 1o)
51		notnot 619	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) = 1o → ¬ ¬ (𝑓‘𝑥) = 1o)
52	51	adantl 275	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = 1o) → ¬ ¬ (𝑓‘𝑥) = 1o)
53	50, 52	2falsed 692	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = 1o) → ((1o ∖ (𝑓‘𝑥)) = 1o ↔ ¬ (𝑓‘𝑥) = 1o))
54	14	ffvelrnda 5620	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ 2o)
55		df2o3 6398	. . . . . . . . . . 11 ⊢ 2o = {∅, 1o}
56	54, 55	eleqtrdi 2259	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ {∅, 1o})
57		elpri 3599	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ {∅, 1o} → ((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) = 1o))
58	56, 57	syl 14	. . . . . . . . 9 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((𝑓‘𝑥) = ∅ ∨ (𝑓‘𝑥) = 1o))
59	43, 53, 58	mpjaodan 788	. . . . . . . 8 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((1o ∖ (𝑓‘𝑥)) = 1o ↔ ¬ (𝑓‘𝑥) = 1o))
60	33, 59	bitrd 187	. . . . . . 7 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = 1o ↔ ¬ (𝑓‘𝑥) = 1o))
61	60	ralbidva 2462	. . . . . 6 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → (∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 ¬ (𝑓‘𝑥) = 1o))
62	61	notbid 657	. . . . 5 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → (¬ ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = 1o ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ (𝑓‘𝑥) = 1o))
63		ralnex 2454	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝑓‘𝑥) = 1o ↔ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
64	63	notbii 658	. . . . 5 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ (𝑓‘𝑥) = 1o ↔ ¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
65	62, 64	bitrdi 195	. . . 4 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → (¬ ∀𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = 1o ↔ ¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
66	32	eqeq1d 2174	. . . . . 6 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = ∅ ↔ (1o ∖ (𝑓‘𝑥)) = ∅))
67	35	eqeq1i 2173	. . . . . . . . . . 11 ⊢ ((1o ∖ ∅) = ∅ ↔ 1o = ∅)
68	38, 67	nemtbir 2425	. . . . . . . . . 10 ⊢ ¬ (1o ∖ ∅) = ∅
69	34	eqeq1d 2174	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) = ∅ → ((1o ∖ (𝑓‘𝑥)) = ∅ ↔ (1o ∖ ∅) = ∅))
70	68, 69	mtbiri 665	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) = ∅ → ¬ (1o ∖ (𝑓‘𝑥)) = ∅)
71	70	adantl 275	. . . . . . . 8 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = ∅) → ¬ (1o ∖ (𝑓‘𝑥)) = ∅)
72	71, 42	2falsed 692	. . . . . . 7 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = ∅) → ((1o ∖ (𝑓‘𝑥)) = ∅ ↔ (𝑓‘𝑥) = 1o))
73	47, 44	eqtrdi 2215	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) = 1o → (1o ∖ (𝑓‘𝑥)) = ∅)
74	73	adantl 275	. . . . . . . 8 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = 1o) → (1o ∖ (𝑓‘𝑥)) = ∅)
75		simpr 109	. . . . . . . 8 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = 1o) → (𝑓‘𝑥) = 1o)
76	74, 75	2thd 174	. . . . . . 7 ⊢ ((((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑓‘𝑥) = 1o) → ((1o ∖ (𝑓‘𝑥)) = ∅ ↔ (𝑓‘𝑥) = 1o))
77	72, 76, 58	mpjaodan 788	. . . . . 6 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((1o ∖ (𝑓‘𝑥)) = ∅ ↔ (𝑓‘𝑥) = 1o))
78	66, 77	bitrd 187	. . . . 5 ⊢ (((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = ∅ ↔ (𝑓‘𝑥) = 1o))
79	78	rexbidva 2463	. . . 4 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → (∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑓‘𝑧)))‘𝑥) = ∅ ↔ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
80	24, 65, 79	3imtr3d 201	. . 3 ⊢ ((𝐴 ∈ Markov ∧ 𝑓 ∈ (2o ↑𝑚 𝐴)) → (¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
81	80	ralrimiva 2539	. 2 ⊢ (𝐴 ∈ Markov → ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
82		elex 2737	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
83	82	adantr 274	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → 𝐴 ∈ V)
84		fveq1 5485	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))) → (𝑓‘𝑥) = ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥))
85	84	eqeq1d 2174	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))) → ((𝑓‘𝑥) = 1o ↔ ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o))
86	85	rexbidv 2467	. . . . . . . . . 10 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))) → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o))
87	86	notbid 657	. . . . . . . . 9 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))) → (¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ¬ ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o))
88	87	notbid 657	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))) → (¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ¬ ¬ ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o))
89	88, 86	imbi12d 233	. . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))) → ((¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ↔ (¬ ¬ ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o)))
90		simplr 520	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
91		elmapi 6636	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (2o ↑𝑚 𝐴) → 𝑔:𝐴⟶2o)
92	91	adantl 275	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → 𝑔:𝐴⟶2o)
93	92	ffvelrnda 5620	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑔‘𝑧) ∈ 2o)
94		2oconcl 6407	. . . . . . . . . 10 ⊢ ((𝑔‘𝑧) ∈ 2o → (1o ∖ (𝑔‘𝑧)) ∈ 2o)
95	93, 94	syl 14	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑧 ∈ 𝐴) → (1o ∖ (𝑔‘𝑧)) ∈ 2o)
96	95	fmpttd 5640	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))):𝐴⟶2o)
97	19	a1i 9	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → 2o ∈ ω)
98		simpll 519	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → 𝐴 ∈ 𝑉)
99	97, 98	elmapd 6628	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))) ∈ (2o ↑𝑚 𝐴) ↔ (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))):𝐴⟶2o))
100	96, 99	mpbird 166	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))) ∈ (2o ↑𝑚 𝐴))
101	89, 90, 100	rspcdva 2835	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (¬ ¬ ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o))
102		ralnex 2454	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ¬ ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o ↔ ¬ ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o)
103	102	notbii 658	. . . . . . 7 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o ↔ ¬ ¬ ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o)
104		nfv 1516	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
105		nfcv 2308	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(2o ↑𝑚 𝐴)
106		nfre1 2509	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o
107	106	nfn 1646	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o
108	107	nfn 1646	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 ¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o
109	108, 106	nfim 1560	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
110	105, 109	nfralxy 2504	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)
111	104, 110	nfan 1553	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
112		nfv 1516	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑔 ∈ (2o ↑𝑚 𝐴)
113	111, 112	nfan 1553	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴))
114		eqid 2165	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧))) = (𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))
115		fveq2 5486	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑔‘𝑧) = (𝑔‘𝑥))
116	115	difeq2d 3240	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (1o ∖ (𝑔‘𝑧)) = (1o ∖ (𝑔‘𝑥)))
117		simpr 109	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
118		difexg 4123	. . . . . . . . . . . . . 14 ⊢ (1o ∈ V → (1o ∖ (𝑔‘𝑥)) ∈ V)
119	29, 118	mp1i 10	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (1o ∖ (𝑔‘𝑥)) ∈ V)
120	114, 116, 117, 119	fvmptd3 5579	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = (1o ∖ (𝑔‘𝑥)))
121	120	eqeq1d 2174	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o ↔ (1o ∖ (𝑔‘𝑥)) = 1o))
122	121	notbid 657	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (¬ ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o ↔ ¬ (1o ∖ (𝑔‘𝑥)) = 1o))
123		difeq2 3234	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘𝑥) = ∅ → (1o ∖ (𝑔‘𝑥)) = (1o ∖ ∅))
124	123, 35	eqtrdi 2215	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑥) = ∅ → (1o ∖ (𝑔‘𝑥)) = 1o)
125	124	adantl 275	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = ∅) → (1o ∖ (𝑔‘𝑥)) = 1o)
126	125	notnotd 620	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = ∅) → ¬ ¬ (1o ∖ (𝑔‘𝑥)) = 1o)
127		eqeq1 2172	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑥) = ∅ → ((𝑔‘𝑥) = 1o ↔ ∅ = 1o))
128	39, 127	mtbiri 665	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑥) = ∅ → ¬ (𝑔‘𝑥) = 1o)
129	128	adantl 275	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = ∅) → ¬ (𝑔‘𝑥) = 1o)
130	126, 129	2falsed 692	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = ∅) → (¬ (1o ∖ (𝑔‘𝑥)) = 1o ↔ (𝑔‘𝑥) = 1o))
131		difeq2 3234	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘𝑥) = 1o → (1o ∖ (𝑔‘𝑥)) = (1o ∖ 1o))
132	131	eqeq1d 2174	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑥) = 1o → ((1o ∖ (𝑔‘𝑥)) = 1o ↔ (1o ∖ 1o) = 1o))
133	46, 132	mtbiri 665	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑥) = 1o → ¬ (1o ∖ (𝑔‘𝑥)) = 1o)
134	133	adantl 275	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = 1o) → ¬ (1o ∖ (𝑔‘𝑥)) = 1o)
135		simpr 109	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = 1o) → (𝑔‘𝑥) = 1o)
136	134, 135	2thd 174	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = 1o) → (¬ (1o ∖ (𝑔‘𝑥)) = 1o ↔ (𝑔‘𝑥) = 1o))
137	91	ad2antlr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑔:𝐴⟶2o)
138	137, 117	ffvelrnd 5621	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ 2o)
139	138, 55	eleqtrdi 2259	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ {∅, 1o})
140		elpri 3599	. . . . . . . . . . . 12 ⊢ ((𝑔‘𝑥) ∈ {∅, 1o} → ((𝑔‘𝑥) = ∅ ∨ (𝑔‘𝑥) = 1o))
141	139, 140	syl 14	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((𝑔‘𝑥) = ∅ ∨ (𝑔‘𝑥) = 1o))
142	130, 136, 141	mpjaodan 788	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (¬ (1o ∖ (𝑔‘𝑥)) = 1o ↔ (𝑔‘𝑥) = 1o))
143	122, 142	bitrd 187	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (¬ ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o ↔ (𝑔‘𝑥) = 1o))
144	113, 143	ralbida 2460	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (∀𝑥 ∈ 𝐴 ¬ ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o ↔ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o))
145	144	notbid 657	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (¬ ∀𝑥 ∈ 𝐴 ¬ ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o ↔ ¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o))
146	103, 145	bitr3id 193	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (¬ ¬ ∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o ↔ ¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o))
147		simpr 109	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = ∅) → (𝑔‘𝑥) = ∅)
148	125, 147	2thd 174	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = ∅) → ((1o ∖ (𝑔‘𝑥)) = 1o ↔ (𝑔‘𝑥) = ∅))
149	128, 135	nsyl3 616	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = 1o) → ¬ (𝑔‘𝑥) = ∅)
150	134, 149	2falsed 692	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑔‘𝑥) = 1o) → ((1o ∖ (𝑔‘𝑥)) = 1o ↔ (𝑔‘𝑥) = ∅))
151	148, 150, 141	mpjaodan 788	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → ((1o ∖ (𝑔‘𝑥)) = 1o ↔ (𝑔‘𝑥) = ∅))
152	121, 151	bitrd 187	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o ↔ (𝑔‘𝑥) = ∅))
153	113, 152	rexbida 2461	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (∃𝑥 ∈ 𝐴 ((𝑧 ∈ 𝐴 ↦ (1o ∖ (𝑔‘𝑧)))‘𝑥) = 1o ↔ ∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅))
154	101, 146, 153	3imtr3d 201	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) ∧ 𝑔 ∈ (2o ↑𝑚 𝐴)) → (¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅))
155	154	ralrimiva 2539	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → ∀𝑔 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅))
156	9	biimprd 157	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑔 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑔‘𝑥) = ∅) → 𝐴 ∈ Markov))
157	83, 155, 156	sylc 62	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)) → 𝐴 ∈ Markov)
158	157	ex 114	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) → 𝐴 ∈ Markov))
159	81, 158	impbid2 142	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  Vcvv 2726   ∖ cdif 3113  ∅c0 3409  {cpr 3577   ↦ cmpt 4043  ωcom 4567  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842  1_oc1o 6377  2_oc2o 6378   ↑_𝑚 cmap 6614  Markovcmarkov 7115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1o 6384  df-2o 6385  df-map 6616  df-markov 7116

This theorem is referenced by:  subctctexmid  13881